
\(\beta \) -transformations.
The \(\beta \) -transformation \(T_{\beta }\colon [0,1]\rightarrow [0,1]\)  with \(\beta >1\)  was introduced by Rényi [1]} and defined by
\(T_{\beta }(x)=\left\lbrace \begin{array}{ll}\beta x\ (\text{mod}\ 1) & (x\ne 1),\\\displaystyle \lim _{y\rightarrow 1-0}(\beta y\ (\text{mod}\ 1)) & (x=1).\end{array}\right.\) 


